Explicit schemes for dam-break simulations

Abstract

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.